So i have this question:
To formulate a zero sum game, these are my assumptions:
- B,C,D=1,1,1 and a > 0.
-p1 is the guard and p2 is the burglar.
-p1 goes on the rows and p2 goes on the columns.
-if p1 wins, he gets a payoff of a , and if p2 looses he gets a payoff of -a.
- ABCD are the corresponding row labels and column labels..
First of all, are these assumptions correct?
Then i wanted to analyze the first column.
If P1 is in A, and P2 is in A, then P1 gets an outcome of a?
If P1 is in B, and p2 is in A, then P1 gets an outcome of -a?
If my ideas are correct, then the matrix represents the payoffs of P1, the guard, instead of the payoffs of the burglar. Then if P1 doesn't win, his payoff should be -a too if he is in the a room.
Should this be right?
Moroever, why are the a values only in the first column? For example, in the first row second column, since player 1 is in a but player 2 is in b, shouldn't the value still be -a?
Am i doing something wrong?
The third of your interpretations/assumptions is wrong. It should be: "If $P1$ wins then $P1$ receives $a$ if $P2$ is in $A$, otherwise he receives $1$." The entries in the matrix can be thought of as the payout for $P1$. Negative entries correspond to cases where $P1$ looses. In those cases the entries represent the negative of what $P2$ wins.
If $P1$ is in $A$ and $P2$ in $A$ then the thieve is caught resulting in a payout of $a$ for $P1$. If $P1$ is in $B$ and $P2$ in $A$ then the thieve is not caught as $A$ is not adjacent to $B$. Hence $P2$ gets payout of $a$ and this is represented by the value $-a$ in the matrix.
In the second to fourth column the values are either $1$ or $-1$ because in those cases the burglar finds himself in one of the rooms $B, C, D$. In those rooms objects have value $1$ and thus either the guard or the burglar get payout $1$.